Bài 15 trang 233 SBT đại số và giải tích 11

Tính giới hạn \(\mathop {\lim }\limits_{n \to  + \infty } {x_n}\)

LG a

\({x_n} = \frac{{\sqrt {{n^2} + 1}  + \sqrt n }}{{\sqrt[3]{{{n^3} + n}} - n}}\)

Lời giải chi tiết:

\(\begin{array}{l}\lim {x_n} = \lim \frac{{\sqrt {{n^2} + 1}  + \sqrt n }}{{\sqrt[3]{{{n^3} + n}} - n}}\\ = \lim \frac{{\sqrt {{n^2}\left( {1 + \frac{1}{{{n^2}}}} \right)}  + \frac{n}{{\sqrt n }}}}{{\sqrt[3]{{{n^3}\left( {1 + \frac{1}{{{n^2}}}} \right)}} - n}}\\ = \lim \frac{{n\sqrt {1 + \frac{1}{{{n^2}}}}  + n.\frac{1}{{\sqrt n }}}}{{n\sqrt[3]{{1 + \frac{1}{{{n^2}}}}} - n}}\\ = \lim \frac{{n\left( {\sqrt {1 + \frac{1}{{{n^2}}}}  + \frac{1}{{\sqrt n }}} \right)}}{{n\left( {\sqrt[3]{{1 + \frac{1}{{{n^2}}}}} - 1} \right)}}\\ = \lim \frac{{\sqrt {1 + \frac{1}{{{n^2}}}}  + \frac{1}{{\sqrt n }}}}{{\sqrt[3]{{1 + \frac{1}{{{n^2}}}}} - 1}} =  + \infty \end{array}\)

Vì \(\lim \left( {\sqrt {1 + \frac{1}{{{n^2}}}}  + \frac{1}{{\sqrt n }}} \right) = 1 > 0\) và \(\left\{ \begin{array}{l}\lim \left( {\sqrt[3]{{1 + \frac{1}{{{n^2}}}}} - 1} \right) = 0\\\sqrt[3]{{1 + \frac{1}{{{n^2}}}}} - 1 > 0\end{array} \right.\)

LG b

\({x_n} = \left( {n - \frac{1}{n}} \right)\left( {\frac{{1 - 4n}}{{2{n^2}}}} \right)\)

Lời giải chi tiết:

\(\begin{array}{l}\lim {x_n} = \lim \left( {n - \frac{1}{n}} \right)\left( {\frac{{1 - 4n}}{{2{n^2}}}} \right)\\ = \lim \frac{{\left( {{n^2} - 1} \right)\left( {1 - 4n} \right)}}{{2{n^3}}}\\ = \lim \frac{{\frac{{{n^2} - 1}}{{{n^2}}}.\frac{{1 - 4n}}{n}}}{{\frac{{2{n^3}}}{{{n^3}}}}}\\ = \lim \frac{{\left( {1 - \frac{1}{{{n^2}}}} \right).\left( {\frac{1}{n} - 4} \right)}}{2}\\ = \frac{{\left( {1 - 0} \right)\left( {0 - 4} \right)}}{2} =  - 2\end{array}\)

Cách khác:

\(\begin{array}{l}\lim {x_n} = \lim \left( {n - \frac{1}{n}} \right)\left( {\frac{{1 - 4n}}{{2{n^2}}}} \right)\\ = \lim \left[ {n\left( {1 - \frac{1}{{{n^2}}}} \right).\frac{1}{n}\left( {\frac{{1 - 4n}}{{2n}}} \right)} \right]\\ = \lim \left[ {n\left( {1 - \frac{1}{{{n^2}}}} \right).\frac{1}{n}\left( {\frac{1}{{2n}} - 2} \right)} \right]\\ = \lim \left( {1 - \frac{1}{{{n^2}}}} \right)\left( {\frac{1}{{2n}} - 2} \right)\\ = \left( {1 - 0} \right)\left( {0 - 2} \right) =  - 2\end{array}\)